Time-dependent DFT offers a very appealing solution to these problems. Recall
that TDDFT equations simulate a complete fission event while, at the same time,
conserving the total energy. If we wait long enough when running the
simulation, we will reach the point where the two fragments are well separated
with only the Coulomb force acting between them; see Fig. REF . The
energy balance reads \(E_0 = E_{\rm H}(t) + E_{\rm L}(t) + E_{\rm int}(t)\) ,
where \(E_0 = E(t=0)\)  is the total, conserved energy of the system,
\(E_{\rm H,L}(t)\)  is the time-dependent energy of the heavy/light fragment and
\(E_{\rm int}(t)\)  is the interaction energy between them. At late times, or
large distances between the two fragments, this interaction energy reduces to
the total kinetic energy which is easily obtained from TDDFT evolution
[1]}, [2]}. From that same evolution, we can extract the
density matrix \(\rho _{\rm H,L}(t)\)  and pairing tensor \(\kappa _{\rm H,L}(t)\)  of
each fragment and therefore calculate their total energy directly. Subtracting
the ground-state binding energy of each fragment gives the excitation energy
\(E^*{\rm H,L}\) . Estimates of both TKE and the \(E^*{\rm H,L}\)  have been obtained
for the case of neutron-induced fission of \(^{239}\) Pu and agree rather well
with experimental measurements [3]}, [4]}. They also suggest
that, at least for the most likely fission, the heavy fragment tends to have a
lower internal temperature than the light one [4]}, [6]}.
Note that the shape relaxation of the fragments and the large Coulomb
interaction at scission complicate this picture since they lead to the
excitation of giant resonances, which in turn imply that the energy of each
fragment is not constant over time [1]}, [6]}.
